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tangent circles , given equal angles (V Soros Olympiad 1998-99 Round 1 9.10)

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May 25, 2024
geometrytangent circles

Problem Statement

On the bisector of angle AA of triangle ABCABC, points DD and FF are taken inside the triangle so that ∠DBC=∠FBA\angle DBC = \angle FBA. Prove that: a) ∠DCB=∠FCA\angle DCB = \angle FCA b) a circle passing through BB and FF and tangent to the segment BCBC is tangle to the circumscribed circle of the triangle ABCABC.
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